aerodynamic inverse design and shape optimization via control

Aerodynamic Inverse Designand

Shape Optimization via Control Theory

Antony Jameson1

1Department of Aeronautics & AstronauticsStanford University

University of Texas, AustinFebruary 20, 2015

1 IntroductionIntroductionLighthill’s MethodControl Theory Approach to Design

2 Numerical FormulationDesign using the Euler EquationsSobolev Inner Product

3 Design ProcessDesign Process OutlineDrag MinimizationInverse DesignP51 RacerReduced Sweep: Design for the CRMLow Sweep Wing DesignNatural Laminar WingNumerical Wind Tunnel

4 AppendixDesign using the Transonic Potential Flow Equation

Introduction Numerical Formulation Design Process Appendix

INTRODUCTION

Antony Jameson Mathematics of Aerodynamic Shape Optimization


Objective of Computational Aerodynamics

1 Capability to predict the flow past an airplane in different flight regimes such astake off, cruise, flutter.

2 Interactive design calculations to allow immediate improvement3 Automatic design optimization



Early Aerodynamic Design Methods

1945 Lighthill (Conformal Mapping, Incompressible Flow)

1965 Nieuwland (Hodograph, Power Series)

1970 Garabedian - Korn (Hodograph, Complex Characteristics)

1974 Boerstoel (Hodograph)

1974 Trenen (Potential Flow, Dirichlet Boundary Conditions)

1977 Henne (3D Potential Flow, based on FLO22)

1985 Volpe-Melnik (2D Potential Flow, Bsed on FLO36)

1979 Garabedian-McFadden (Potential Flow, Neumann Boundary Conditions,Iterated Mapping)

1976 Sobieczi (Fictitious Gas)

1979 Drela-Giles (2D Euler Equations, Streamline Coordinates, Newton Iteration)



LIGHTHILL’S METHOD



Lighthill’s Method

Profile P on z plane

⇓Profile C on σ plane

Let the profile P be conformally mapped to anunit circle C

The surface velocity is q = 1h|∇φ| where φ is

the potential in the circle plane, and h is themapping modulus h =

˛dzdσ

˛= ds

dθ

Choose q = qT

Solve for the maping modulus h = 1qT|∇φ|



Implementation of Lighthill’s Method

Design Profile C for Specified Surface speed qt. Let a profile C be conformally mapped to a circle by

logdz

dσ=X Cn

σn

logds

dθ+ i(α− θ −

pi

2) =

X(ancos(nθ) + bnsin(nθ)) + i

X(bncos(nθ)− ansin(nθ))

where

q =∇Φ

h, h =

˛dz

dσ

˛and

Φ = (r +1

r)cosθ +

Γ

2πθ is known

On C set q = qt

→ds

dθ=

Φθ

qt→ an, bb



Constraints with Lighthill’s Method

To preserve q∞c0 = 0

Also, integration around a circuit gives

∆z =

Idz

dσdσ = 2πic1

Closure→ c1 = 0Thus, Z

log(qt)dθ = 0

Zlog(qt)cos(θ)dθ = 0

Zlog(qt)sin(θ)dθ = 0



CONTROL THEORY APPROACH TO DESIGN



Control Theory Approach to Design

A wing is a device to control the flow. Apply the theory of control ofpartial differential equations (J.L.Lions) in conjunction with CFD.

References

Pironneau (1964) Optimum shape design for subsonic potential flow

Jameson (1988) Optimum shape design for transonic and supersonic flowmodeled by the transonic potential flow equation and the Euler equations



Control Theory Approach to the Design Method

Define a cost functionI =

1

2

ZB

(p− pt)2dB

orI =

1

2

ZB

(q − qt)2dB

The surface shape is now treated as the control, which is to be varied to minimize I,subject to the constraint that the flow equations are satisfied in the domain D.



Choice of Domain

ALTERNATIVES1 Variable computational domain - Free boundary problem2 Transformation to a fixed computational domain - Control via the transformation

function

EXAMPLES1 2D via Conformal mapping with potential flow2 2D via Conformal mapping with Euler equations3 3D Sheared Parabolic Coordinates with Euler equation4 ...



Formulation of the Control Problem

Suppose that the surface of the body is expressed by an equation

f(x) = 0

Vary f to f + δf and find δI.If we can express

δI =

ZBgδfdB = (g, δf)B

Then we can recognize g as the gradient ∂I∂f

.Choose a modification

δf = −λg

Then to first orderδI = −λ(g, g)B ≤ 0

In the presence of constraints project g into the admissible trial space.

Accelerate by the conjugate gradient method.



Traditional Approach to Design Optimization

Define the geometry through a set of design parameters, for example, to be the weights αi applied to a set of shape functionsbi(x) so that the shape is represented as

f(x) =X

αibi(x).

Then a cost function I is selected , for example, to be the drag coefficient or the lift to drag ratio, and I is regarded as a functionof the parameters αi. The sensitivities ∂I

∂αimay be estimated by making a small variation δαi in each design parameter in turn

and recalculating the flow to obtain the change in I. Then

∂I

∂αi≈I(αi + δαi)− I(αi)

δαi.

The gradient vector G = ∂I∂α

may now be used to determine a direction of improvement. The simplest procedure is to make astep in the negative gradient direction by setting

αn+1

= αn

+ δα,

whereδα = −λG

so that to first orderI + δI = I − GT δα = I − λGT G < I



Disadvantages

The main disadvantage of this approach is the need for a number of flow calculationsproportional to the number of design variables to estimate the gradient. Thecomputational costs can thus become prohibitive as the number of design variables isincreased.



Formulation of the Adjoint Approach to Optimal Design

For flow about an airfoil or wing, the aerodynamic properties which define the cost function are functions of the flow-fieldvariables (w) and the physical location of the boundary, which may be represented by the function F , say. Then

I = I (w,F) ,

and a change in F results in a change

δI =

"∂IT

∂w

#δw +

"∂IT

∂F

#δF (1)

in the cost function. Suppose that the governing equation R which expresses the dependence of w and F within the flowfielddomain D can be written as

R (w,F) = 0. (2)

Then δw is determined from the equation

δR =

»∂R

∂w

–δw +

»∂R

∂F

–δF = 0. (3)

Since the variation δR is zero, it can be multiplied by a Lagrange Multiplier ψ and subtracted from the variation δI withoutchanging the result.



Formulation of the Adjoint Approach to Optimal Design

δI =∂IT

∂wδw +

∂IT

∂FδF − ψT

„»∂R

∂w

–δw +

»∂R

∂F

–δF«

=

(∂IT

∂w− ψT

»∂R

∂w

–)δw +

(∂IT

∂F− ψT

»∂R

∂F

–)δF. (4)

Choosing ψ to satisfy the adjoint equation »∂R

∂w

–Tψ =

∂I

∂w(5)

the first term is eliminated, and we find thatδI = GT δF, (6)

where

G =∂IT

∂F− ψT

»∂R

∂F

–.

An improvement can be made with a shape changeδF = −λG

where λ is positive and small enough that the first variation is an accurate estimate of δI.



Advantages

The advantage is that (6) is independent of δw, with the result that the gradient of I with respect to an arbitrary number ofdesign variables can be determined without the need for additional flow-field evaluations.

The cost of solving the adjoint equation is comparable to that of solving the flow equations. Thus the gradient can bedetermined with roughly the computational costs of two flow solutions, independently of the number of design variables,which may be infinite if the boundary is regarded as a free surface.

When the number of design variables becomes large, the computational efficiency of the control theory approach overtraditional approach, which requires direct evaluation of the gradients by individually varying each design variable andrecomputing the flow fields, becomes compelling.



DESIGN USING THE EULER EQUATIONS



Design using the Euler Equations

In a fixed computational domain with coordinates, ξ, the Euler equations are

J∂w

∂t+ R(w) = 0 (7)

where J is the Jacobian (cell volume),

R(w) =∂

∂ξi(Sijfj) =

∂Fi

∂ξi. (8)

and Sij are the metric coefficients (face normals in a finite volume scheme). We can write the fluxes in termsof the scaled contravariant velocity components

Ui = Sijuj

as

Fi = Sijfj =

26664ρUi

ρUiu1 + Si1pρUiu2 + Si2pρUiu3 + Si3p

ρUiH

37775 .where p = (γ − 1)ρ(E − 1

2u2i ) and ρH = ρE + p.




A variation in the geometry now appears as a variation δSij in the metric coefficients.The variation in the residual is

δR =∂

∂ξi(δSijfj) +

∂

∂ξi

„Sij

∂fj∂w

δw

«(9)

and the variation in the cost δI is augmented as

δI −ZD

ψT δR dξ (10)

which is integrated by parts to yield

δI −ZBψTniδFidξB +

ZD

∂ψT

∂ξ(δSijfj) dξ +

ZD

∂ψT

∂ξiSij

∂fj∂w

δwdξ




For simplicity, it will be assumed that the portion of the boundary that undergoes shape modifications isrestricted to the coordinate surface ξ2 = 0. Then equations for the variation of the cost function and theadjoint boundary conditions may be simplified by incorporating the conditions

n1 = n3 = 0, n2 = 1, Bξ = dξ1dξ3,

so that only the variation δF2 needs to be considered at the wall boundary. The condition that there is no flowthrough the wall boundary at ξ2 = 0 is equivalent to

U2 = 0, so that δU2 = 0

when the boundary shape is modified. Consequently the variation of the inviscid flux at the boundary reducesto

δF2 = δp

8>>><>>>:0S21S22S230

9>>>=>>>;+ p

8>>><>>>:0

δS21δS22δS23

0

9>>>=>>>; . (11)




In order to design a shape which will lead to a desired pressure distribution, a naturalchoice is to set

I =1

2

ZB

(p− pd)2 dS

where pd is the desired surface pressure, and the integral is evaluated over the actualsurface area. In the computational domain this is transformed to

I =1

2

ZZBw

(p− pd)2 |S2| dξ1dξ3,

where the quantity|S2| =

pS2jS2j

denotes the face area corresponding to a unit element of face area in thecomputational domain.




In the computational domain the adjoint equation assumes the form

CTi∂ψ

∂ξi= 0 (12)

whereCi = Sij

∂fj∂w

.

To cancel the dependence of the boundary integral on δp, the adjoint boundarycondition reduces to

ψjnj = p− pd (13)

where nj are the components of the surface normal

nj =S2j

|S2|.




This amounts to a transpiration boundary condition on the co-state variables corresponding to the momentumcomponents. Note that it imposes no restriction on the tangential component of ψ at the boundary.We find finally that

δI = −ZD

∂ψT

∂ξiδSijfjdD

−ZZ

BW

(δS21ψ2 + δS22ψ3 + δS23ψ4) p dξ1dξ3. (14)

Here the expression for the cost variation depends on the mesh variations throughout the domain which appear

in the field integral. However, the true gradient for a shape variation should not depend on the way in which the

mesh is deformed, but only on the true flow solution. In the next section we show how the field integral can be

eliminated to produce a reduced gradient formula which depends only on the boundary movement.



The Reduced Gradient Formulation

Consider the case of a mesh variation with a fixed boundary. Then δI = 0 but there is a variation in thetransformed flux,

δFi = Ciδw + δSijfj .

Here the true solution is unchanged. Thus, the variation δw is due to the mesh movement δx at each meshpoint. Therefore

δw = ∇w · δx =∂w

∂xjδxj

`= δw

∗´and since ∂

∂ξiδFi = 0, it follows that

∂

∂ξi(δSijfj) = −

∂

∂ξi

`Ciδw

∗´. (15)

It has been verified by Jameson and KimF that this relation holds in the general case with boundarymovement.F “Reduction of the Adjoint Gradient Formula in the Continuous Limit”, A.Jameson and S. Kim, 41st AIAA Aerospace Sciences Meeting & Exhibit, AIAA Paper 2003–0040, Reno, NV, January 6–9, 2003.



The Reduced Gradient Formulation

Now ZDφTδR dD =

ZDφT ∂

∂ξiCi`δw − δw∗

´dD

=

ZBφTCi`δw − δw∗

´dB

−ZD

∂φT

∂ξiCi`δw − δw∗

´dD. (16)

Here on the wall boundaryC2δw = δF2 − δS2jfj . (17)

Thus, by choosing φ to satisfy the adjoint equation and the adjoint boundary condition, we reduce the cost variation to aboundary integral which depends only on the surface displacement:

δI =

ZBW

ψT `

δS2jfj + C2δw∗´dξ1dξ3

−ZZBW

(δS21ψ2 + δS22ψ3 + δS23ψ4) p dξ1dξ3. (18)



SOBOLEV INNER PRODUCT



The Need for a Sobolev Inner Product in the Definition of the Gradient

Another key issue for successful implementation of the continuous adjoint method is the choice of an appropriate inner productfor the definition of the gradient. It turns out that there is an enormous benefit from the use of a modified Sobolev gradient, whichenables the generation of a sequence of smooth shapes. This can be illustrated by considering the simplest case of a problem inthe calculus of variations.Suppose that we wish to find the path y(x) which minimizes

I =

Z baF (y, y

′)dx

with fixed end points y(a) and y(b).Under a variation δy(x),

δI =

Z b1

„∂F

∂yδy +

∂F

∂y′δy′«dx

=

Z b1

„∂F

∂y−

d

dx

∂F

∂y′

«δydx




Thus defining the gradient as

g =∂F

∂y−

d

dx

∂F

∂y′

and the inner product as

(u, v) =

Z bauvdx

we find thatδI = (g, δy).

If we now setδy = −λg, λ > 0,

we obtain a improvementδI = −λ(g, g) ≤ 0

unless g = 0, the necessary condition for a minimum.




Note that g is a function of y, y′, y′′,g = g(y, y

′, y′′

)

In the well known case of the Brachistrone problem, for example, which calls for the determination of the path of quickest descentbetween two laterally separated points when a particle falls under gravity,

F (y, y′) =

vuut 1 + y′2

y

and

g = −1 + y

′2+ 2yy′′

2hy(1 + y

′2)i3/2

It can be seen that each stepyn+1

= yn − λngn

reduces the smoothness of y by two classes. Thus the computed trajectory becomes less and less smooth, leading to instability.




In order to prevent this we can introduce a weighted Sobolev inner product

〈u, v〉 =

Z(uv + εu

′v′)dx

where ε is a parameter that controls the weight of the derivatives. We now define a gradient g such thatδI = 〈g, δy〉. Then we have

δI =

Z(gδy + εg

′δy′)dx

=

Z „g −

∂

∂xε∂g

∂x

«δydx

= (g, δy)

where

g −∂

∂xε∂g

∂x= g

and g = 0 at the end points.




Therefore g can be obtained from g by a smoothing equation.Now the step

yn+1 = yn − λngn

gives an improvementδI = −λn〈gn, gn〉

but yn+1 has the same smoothness as yn, resulting in a stable process.



OUTLINE OF THE DESIGN PROCESS



Outline of the Design Process

The design procedure can finally be summarized as follows:1 Solve the flow equations for ρ, u1, u2, u3 and p.2 Solve the adjoint equations for ψ subject to appropriate boundary conditions.3 Evaluate G and calculate the corresponding Sobolev gradient G.4 Project G into an allowable subspace that satisfies any geometric constraints.5 Update the shape based on the direction of steepest descent.6 Return to 1 until convergence is reached.



Design Cycle

Sobolev Gradient

Gradient Calculation

Flow Solution

Adjoint Solution

Shape & Grid

Repeat the Design Cycleuntil Convergence

Modification



Constraints

Fixed CL.

Fixed span load distribution to present too large CL on the outboard wing whichcan lower the buffet margin.Fixed wing thickness to prevent an increase in structure weight.

Design changes can be can be limited to a specific spanwise range of the wing.Section changes can be limited to a specific chordwise range.

Smooth curvature variations via the use of Sobolev gradient.



Application of Thickness Constraints

Prevent shape change penetrating a specified skeleton (colored in red).

Separate thickness and camber allow free camber variations.

Minimal user input needed.



Computational CostF

Cost of Search Algorithm

Steepest Descent O(N2) StepsQuasi-Newton O(N) StepsSmoothed Gradient O(K) Steps

Note: K is independent of N .

F: “Studies of Alternative Numerical Optimization Methods Applied to the Brachistrone Problem”,A.Jameson and J. Vassberg, Computational Fluid Dynamics, Journal, Vol. 9, No.3, Oct. 2000, pp. 281-296



Computational CostF

Total Computational Cost of Design

+ Finite difference gradients O(N3)Steepest descent

+ Finite difference gradients O(N2)Quasi-Newton

+ Adjoint gradients O(N)Quasi-Newton

+ Adjoint gradients O(K)Smoothed gradientNote: K is independent of N .

F: “Studies of Alternative Numerical Optimization Methods Applied to the Brachistrone Problem”,A.Jameson and J. Vassberg, Computational Fluid Dynamics, Journal, Vol. 9, No.3, Oct. 2000, pp. 281-296



DRAG MINIMIZATION



Viscous Optimization of Korn Airfoil

Initial Final

Mesh size=512 x 64, Mach number=0.75, CLtarget=0.63,Reynolds number=20 Million



The Effect of Applying Smoothed Gradient

Unsmoothed Smoothed



INVERSE DESIGN

Recovering of ONERA M6 Wingfrom its pressure distribution

A. Jameson 2003–2004



NACA 0012 WING TO ONERA M6 TARGET

(a) Staring wing: NACA 0012 (b) Target wing: ONERA M6This is a difficult problem because of the presence of the shock wave in the target pressure and because the

profile to be recovered is symmetric while the target pressure is not.



Pressure Profile at 48% Span

(c) Staring wing: NACA 0012 (d) Target wing: ONERA M6The pressure distribution of the final design match the specified target, even inside the shock.



P51 RACER



P51 Racer

Aircraft competing in the Reno Air Races reach speeds above 500 MPH, encounting compressibility drag due to theappearance of shock waves.

Objective is to delay drag rise without altering the wing structure. Hence try adding a bump on the wing surface.



Partial Redesign

Allow only outward movement.

Limited changes to front part of the chordwise range.



DO WE NEED SWEPT WINGS ON COMMERCIAL JETS?



Background for Studies of Reduced Sweep

Current Transonic TransportsCruise Mach: 0.76 ≤ M ≤ 0.86C/4 Sweep: 25◦ ≤ Λ ≤ 35◦

Wing Planform Layout Knowledge BaseHeavily Influenced By Design ChartsData Developed From Cut-n-Try DesignsData Aumented With Parametric VariationsData Collected Over The YearsIncludes Shifts Due To Technologiese.g., Supercritical Airfoils, Composites, etc.



Pure Aerodynamic Optimizations

Evolution of Pressures for Λ = 10◦ Wing during Optimization



Pure Aerodynamic Optimizations

Mach Sweep CL CD CD.tot ML/D√ML/D

0.85 35◦ 0.500 153.7 293.7 14.47 15.700.84 30◦ 0.510 151.2 291.2 14.71 16.050.83 25◦ 0.515 151.2 291.2 14.68 16.110.82 20◦ 0.520 151.7 291.7 14.62 16.140.81 15◦ 0.525 152.4 292.4 14.54 16.160.80 10◦ 0.530 152.2 292.2 14.51 16.220.79 5◦ 0.535 152.5 292.5 14.45 16.26

CD in counts

CD.tot = CD + 140 counts

Lowest Sweep Favors√ML/D ' 4.0%



Conclusion of Swept Wing Study

An unswept wing at Mach 0.80 offers slightly better range efficiency than a sweptwing at Mach 0.85.

It would also improve TO, climb, descent and landing.

Perhaps B737/A320 replacements should have unswept wings.



DEMO OF LOW SWEEP DESIGN



Application II: Low Sweep Wing Redesign using RK-SGS Scheme

Initial Final

Mesh size=256x64x48, Design Steps=15, Design variables=127x33=4191 surface mesh points



WING DESIGN FOR NATURAL LAMINAR FLOW



Airplane Geometry

X

Y

Z

X

Y

Z

(a) Wing-body Geometry (b) Wing-body Surface Mesh

Mesh size=256x64x48



Airplane Mesh

IAI-NLFP3

GRID 256 X 64 X 48

K = 1

IAI-NLFP3

GRID 256 X 64 X 48

K = 50

Mesh size=256x64x48



Wing Planform and Sectional Profiles

0 0.5 1 1.5 2

!0.8

!0.6

!0.4

!0.2

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7 8

!5

!4

!3

!2

!1

0

1

(a) Cross Sectional Profiles (b) Wing Planform Shape



Design Points: M=0.40, CL=0.4, 0.6, 0.9, 1.0

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(a) M=.40, CL=.40 (b) M=.40, CL=.60

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(c) M=.40, CL=.90 (d) M=.40, CL=1.00

At M = .40, the lower limit of the operating range is set by the appearance of suction peak. This suction peak appears at the

lower surface when CL < .40, and at the upper surface when CL > .90.




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(a) M=.50, CL=.40 (b) M=.50, CL=.60

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<%2--=433(----<>2-343:(93----<.2#=4=?89--------------------------------------

>@ABCD2---3----*@ABEF/62--34(5=:G#3(--------------------------------------------

HIBE2-:5JK-?5K-8;---------------------------------------------------------------

<62--34J89----<E2-343:5:J----<L2#34:;:9-----------------------------------

*MMN-+@0NBMD2--=845O-+@LB#+7/D

<7-P-#:43

<62--=4:83----<E2-343(:3;----<L2#348;5:----------------------------------

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<7-P-#:43

<62--349(8----<E2-3433J=:----<L2#34(=39----------------------------------

)B7-+@0NBMD2--;:45O-+@LB#+7/D

<7-P-#:43

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:%2--;4;3(----:<2-343=8>?----:.2#;4=>35--------------------------------------

<@ABCD2---3----*@ABEF/62--3453=9G#3(--------------------------------------------

HIBE2-=58J->5J-?K---------------------------------------------------------------

:62--349(8----:E2-343=K=3----:L2#34(;5>-----------------------------------

*MMN-+@0NBMD2--;?45O-+@LB#+7/D

:7-P-#=43

:62--;4(?>----:E2-343(>3=----:L2#345;9(----------------------------------

.BE-+@0NBMD2--5;43O-+@LB#+7/D

:7-P-#=43

:62--34K=K----:E2-343;3>5----:L2#34(((3----------------------------------

)B7-+@0NBMD2--K=45O-+@LB#+7/D

:7-P-#=43

(c) M=.50, CL=1.00 (d) M=.50, CL=1.10

At M = .50, the lower limit of the operating range is set by the appearance of suction peak. This suction peak appears at the

lower surface when CL < .40, and at the upper surface when CL > 1.0.




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;7-P-#943

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;7-P-#943

;62--349@(----;F2#34333@>----;L2#34:@?9----------------------------------

)C7-+A0NCME2--<94?O-+ALC#+7/E

;7-P-#943

!"!#$%&'(#)*"$+!)!,$------------------------------------------------./012-34533----"671/2-8439:-----------------------------------------------------

;%2--34533----;<2-3438=>?----;.2#34??=>--------------------------------------

<@ABCD2---3----*@ABEF/62--34899=G#3(--------------------------------------------

HIBE2-:J?K-5JK-=>---------------------------------------------------------------

;62--34=3=----;E2-34385?J----;L2#34:3::-----------------------------------

*MMN-+@0NBMD2--8=4JO-+@LB#+7/D

;7-P-#:43

;62--34939----;E2-343:8(8----;L2#34(>98----------------------------------

.BE-+@0NBMD2--J843O-+@LB#+7/D

;7-P-#:43

;62--34=J9----;E2#34333:9----;L2#34:::9----------------------------------

)B7-+@0NBMD2-->:4JO-+@LB#+7/D

;7-P-#:43

(a) M=.60, CL=.40 (b) M=.60, CL=.60

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9%2--34:3;----9<2-343;=>=----9.2#=43?(?--------------------------------------

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97-P-#;43

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97-P-#;43

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)B7-+@0NBMD2--:;4GO-+@LB#+7/D

97-P-#;43

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<7-P-#:43

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<7-P-#:43

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)A7-+?0NAMC2--F:4>O-+?LA#+7/C

<7-P-#:43

(c) M=.60, CL=.90 (d) M=.60, CL=1.00

At M = .60, the lower limit of the operating range is set by the appearance of suction peak at the lower surface, and the

formation of shock at the upper surface. The suction peak appears at the lower surface when CL < .40, and the shock starts to

form at the upper surface when CL > .90.



Design Points: M=0.65, CL=0.4, 0.5

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:8-P-#=43

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:8-P-#=43

:72--34=?;----:E2#34339(?----:L2#349>35----------------------------------

)B8-+@0NBMD2--G=46O-+@LB#+8/D

:8-P-#=43

!"!#$%&'(#)*"$+!)!,$------------------------------------------------./012-34563----"781/2-3496:-----------------------------------------------------

;%2--34633----;<2-3439=9>----;.2#345?96--------------------------------------

<@ABCD2---3----*@ABEF/72--349?>9G#3(--------------------------------------------

HIBE2-:6JK-56K-=>---------------------------------------------------------------

;72--34(9>----;E2-3439563----;L2#349?96-----------------------------------

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;8-P-#:43

;72--34J33----;E2-3439>:J----;L2#34(J??----------------------------------

.BE-+@0NBMD2--6943O-+@LB#+8/D

;8-P-#:43

;72--34(5J----;E2#34339=3----;L2#34:3==----------------------------------

)B8-+@0NBMD2-->:46O-+@LB#+8/D

;8-P-#:43

(a) M=.65, CL=.40 (b) M=.65, CL=.50

At M = .65, the lower limit of the operating range is set by the appearance of suction peak at the lower surface, and the

formation of shock at the upper surface. The operating range has become too narrow to be useful. In particular, the suction peak

appears at the lower surface when CL <= 40, and the shock of moderate strength has already started to form at the upper

surface when CL = .50.



Operating Envelope (Limit) of Natural Laminar Flows

0.4 0.45 0.5 0.55 0.6 0.650.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

Mach Number

CL

The figure shows the operating boundaries within which favorable pressuredistribution can be maintained.



NUMERICAL WIND TUNNEL



Concept of Numerical Wind Tunnel

FlowSolution

AeroelasticSolution

Loads

Numerically IntensiveHuman Intensive

InitialDesign

CADDefinition

CFDGeometryModeling

MeshGeneration

Requirements Redesign

QuantitativeAssessment

L, D, WMDO*

Visualization

*MDO: Multi-Disciplinary Optimization

1



Advanced Numerical Wind Tunnel

Numerically IntensiveHuman Intensive

InitialDesign

MasterDefinitionCentral

Database

RequirementsHigh-LevelRedesign

GeometryModification

Optimization

QuantitativeAssessment

Automatic MeshGeneration

FlowSolution

Loads

AeroelasticSolution

MonitorResults

1



APPENDIX



DESIGN USING THE TRANSONIC POTENTIAL FLOW EQUATION



Airfoil Design For Potential Flow using Conformal Mapping

Consider the case of two-dimensional compressible inviscid flow. In the absence of shock waves, an initially irrotational flow willremain irrotational, and we can assume that the velocity vector q is the gradient of a potential φ. In the presence of weak shockwaves this remains a fairly good approximation. Let p, ρ, c, and M be the pressure, density, speed-of-sound, and Mach numberq/c. Then the potential flow equation is

∇ · (ρ∇φ) = 0, (19)

where the density is given by

ρ =

1 +

γ − 1

2M

2∞

“1− q2

”ff 1(γ−1)

, (20)

while

p =ργ

γM2∞, c

2=γp

ρ. (21)

HereM∞ is the Mach number in the free stream, and the units have been chosen so that p and q have a value of unity in the far

field.




Suppose that the domain D exterior to the profile C in the z-plane is conformally mapped on to the domain exterior to a unitcircle in the σ-plane. Let R and θ be polar coordinates in the σ-plane, and let r be the inverted radial coordinate 1

R. Also let h

be the modulus of the derivative of the mapping function

h =

˛dz

dσ

˛. (22)

Now the potential flow equation becomes

∂

∂θ(ρφθ) + r

∂

∂r(rρφr) = 0 in D, (23)

where the density is given by equation (20), and the circumferential and radial velocity components are

u =rφθ

h, v =

r2φr

h, (24)

whileq2

= u2

+ v2. (25)




The condition of flow tangency leads to the Neumann boundary condition

v =1

h

∂φ

∂r= 0 on C. (26)

In the far field, the potential is given by an asymptotic estimate, leading to a Dirichlet boundary condition at r = 0.Suppose that it is desired to achieve a specified velocity distribution qd on C. Introduce the cost function

I =1

2

ZC

(q − qd)2dθ,



Design Problem

The design problem is now treated as a control problem where the control function is the mapping modulus h, which is to bechosen to minimize I subject to the constraints defined by the flow equations (19–26).

A modification δh to the mapping modulus will result in variations δφ, δu, δv, and δρ to the potential, velocity components, anddensity. The resulting variation in the cost will be

δI =

ZC

(q − qd) δq dθ, (27)

where, on C, q = u. Also,

δu = rδφθ

h− u

δh

h, δv = r

2 δφr

h− v

δh

h,

while according to equation (20)∂ρ

∂u= −

ρu

c2,∂ρ

∂v= −

ρv

c2.



Design Problem

It follows that δφ satisfies

Lδφ = −∂

∂θ

„ρM

2φθδh

h

«− r

∂

∂r

„ρM

2rφr

δh

h

«where

L ≡∂

∂θ

(ρ

1−

u2

c2

!∂

∂θ−ρuv

c2r∂

∂r

)+ r

∂

∂r

(ρ

1−

v2

c2

!r∂

∂r−ρuv

c2

∂

∂θ

). (28)

Then, if ψ is any periodic differentiable function which vanishes in the far field,ZD

ψ

r2L δφ dS =

ZDρM

2∇φ · ∇ψδh

hdS, (29)

where dS is the area element r dr dθ, and the right hand side has been integrated by parts.



Design Problem

Now we can augment equation (27) by subtracting the constraint (29). The auxiliary function ψ then plays the role of a Lagrangemultiplier. Thus,

δI =

ZC

(q − qd) qδh

hdθ −

ZCδφ

∂

∂θ

„q − qdh

«dθ −

ZD

ψ

r2Lδφ dS +

ZDρM

2∇φ · ∇ψδh

hdS.

Now suppose that ψ satisfies the adjoint equationLψ = 0 in D (30)

with the boundary condition∂ψ

∂r=

1

ρ

∂

∂θ

„q − qdh

«on C. (31)

Then, integrating by parts,

δI = −ZC

(q − qd) qδh

hdθ +

ZDρM

2∇φ · ∇ψδh

hdS. (32)

Here the first term represents the direct effect of the change in the metric, while the area integral represents a correction for the

effect of compressibility. When the second term is deleted the method reduces to a variation of Lighthill’s method.



Design Problem

Equation (32) can be further simplified to represent δI purely as a boundary integral because the mapping function is fullydetermined by the value of its modulus on the boundary. Set

logdz

dσ= F + iβ,

where

F = log

˛dz

dσ

˛= log h,

and

δF =δh

h.

Then F satisfies Laplace’s equation∆F = 0 in D,

and if there is no stretching in the far field, F → 0. Introduce another auxiliary function P which satisfies

∆P = ρM2∇ψ · ∇ψ in D, (33)

andP = 0 on C.



Design Problem

Then after integrating by parts we find that

δI =

ZCG δFc dθ,

where Fc is the boundary value of F , and

G =∂P

∂r− (q − qd) q. (34)

Thus we can attain an improvement by a modification

δFc = −λG

in the modulus of the mapping function on the boundary, which in turn defines the computed mapping function since F satisfies

Laplace’s equation. In this way the Lighthill method is extended to transonic flow.


aerodynamic inverse design and shape optimization via control

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